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Algebraic Fractions

Algebraic Fractions. Remember. Simplifying. e.g. 1. e.g. 2. Ex Simplify. ex 1. ex 2. Ex Simplify. ex 3. ex 4. Multiplying Fractions. e.g 3. e.g 4. Ex Multiply the following. ex 1. ex 2. Dividing Fractions. e.g 5. e.g 6. Ex Divide the following. ex 1. ex 2.

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Algebraic Fractions

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  1. Algebraic Fractions

  2. Remember

  3. Simplifying e.g. 1 e.g. 2

  4. Ex Simplify ex 1 ex 2

  5. Ex Simplify ex 3 ex 4

  6. Multiplying Fractions e.g 3 e.g 4

  7. Ex Multiply the following ex 1 ex 2

  8. Dividing Fractions e.g 5 e.g 6

  9. Ex Divide the following ex 1 ex 2

  10. Common denominators e.g 7 e.g 8 LCM = 6 LCM = ab

  11. Common denominators e.g 9 e.g 10 LCM = 12x2 LCM = 6a2b4

  12. Common denominators e.g 11 e.g 12 LCM = x(x+1) LCM = (x+1)(x-2)

  13. Ex Make a single fraction ex 1 ex 2 LCM = 12 LCM = mn

  14. Ex Make a single fraction ex 3 ex 4 LCM = 8x2 LCM = 15a3b

  15. Ex Make a single fraction ex 5 ex 6 LCM = x(x-3) LCM = (x-4)(x-2)

  16. E.G Fractions and Equations e.g 13 e.g 14 Mult by 2x-1 Mult by 15

  17. E.G Fractions and Equations e.g 15 e.g 16 Mult by x Mult by 6

  18. Ex Solve the following equations ex 1 ex 2 Mult by 2-x Mult by 4

  19. Ex Solve the following Equations ex 3 ex 4 Mult by 2x Mult by 15

  20. Ex Solve the following Equations ex 5 ex 6 Mult by x2 Mult by 10

  21. E.G Making x the subject of formula eg 16 eg 17

  22. E.G Making x the subject of formula eg 18 eg 19

  23. Ex Make x the subject of formula ex 1 ex 2

  24. Ex Make x the subject of formula ex 3 ex 4

  25. Ex Make x the subject of formula ex 5 ex 6

  26. Ratio and Proportion In a Nutshell

  27. The Finding One Method Direct Proportion. Example 1. If it costs 85p for 5 Mars bars, what is the cost of 3 Mars bars ? Solution. ALWAYS Find the cost of 1. This can be easily set out like a table: 5 Mars bars 85 p 5 1 Mars bar 17 p 3 3 Mars bars 51p

  28. Example 2. If 9 blank discs cost £27, how much will 8 blank discs cost? 9 discs £27 9 1 Disc £3 8 8 discs £24.00 Example 3. If Michael can walk 8 km in 2hours, at the same speed how far will he be able to walk in 5hours? 2 hours 8 km 2 1 hour 4 km 5 5 hours 20 km Example 4. If a 300 g bag of chips contains 750 calories, how many calories will be in a 250 g bag? 300 g 750 cal 300 1 g 2.5 cal 250 250 g 625 calories

  29. The Finding One Method Inverse Proportion. Example 1. If it takes 2 men 6 days to build a drive way, how long should it take 3 men? Solution. ALWAYS Find the value of 1. 2 men 6 days 2 2 1 man 12 days 3 3 3 men 4 days

  30. Example 2. If 5men take 12 hrs to fix a road, how long should it take 6 men? 5 men 12 hrs 5 5 1 man 60 hrs 6 6 6 men 10 hrs Example 3. A farmer has enough feed to feed 64 cows for 3 days. How long will the same food last 24 cows? 64 cows 3 days 64 64 1 cow 192 days 24 24 24 cows 8 days Example 4. A journey takes 2hrs travelling at 60 km/h, how long would the journey take travelling at 40 km/h? 60 km/h 2 hrs 60 60 1 km/h 120 hrs 40 40 40 km/h 3 hrs

  31. Graphs Direct proportion Inverse proportion T P n m P  n T  1/m ‘P varies directly as n’ ‘T varies inversely as m’

  32. Worked examples – Direct Proportion e.g. 1 Work, W varies directly as time, t. Given that W = 20 when t = 4 a. Find a formula for W b. Find W when t = 7 e.g. 2 Energy, E varies directly as the square of velocity, v Given that E = 72 when v = 3 a. Find a formula for E b. Find E when v = 4 E  v2 W  t E = k v2 W = k t Sub 72 = 32 k Sub 20 = 4k 72 = 9k k = 5 k = 8 E = 8 v2 W = 5 t E = 8 x 42 W = 5 x 7 E = 8 x 16 W = 35 E = 128

  33. Worked examples – Direct Proportion ex 1 Cost, C varies directly as amount, A. Given that C = 35 when A = 5 a. Find a formula for C b. Find C when A = 9 ex 2 Power, P varies directly as the square of current, I Given that P = 100 when I = 5 a. Find a formula for P b. Find P when I = 3 P  I2 C  A P = k I2 C = k A Sub 100 = 52 k Sub 35 = 5k 100 = 25k k = 7 k = 4 P = 4 I2 C = 7 A P = 4 x 32 C = 7 x 9 P = 4 x 9 C = 63 P = 36

  34. Worked examples – Direct Proportion ex 3 Cost, C varies directly as number, n. Given that C = 36 when n = 3 a. Find a formula for C b. Find C when n = 7 ex 4 D varies directly as the square root of h, Given that d = 30 when h = 9 a. Find a formula for D b. Find D when h = 49 D  h C  n D = k h C = k n Sub 30 = 9 k Sub 36 = 3k 30 = 3k k = 12 k = 10 D = 10 h C = 12 n D = 10 x 49 C = 12 x 7 D = 10 x 7 C = 84 D = 70

  35. Worked examples – Inverse Proportion eg 1 Time, T varies indirectly as the number, n. Given that T = 40 when n = 3 a. Find a formula for T b. Find T when n = 10 eg 2 D varies inversely as the square root of r, Given that D = 5 when r = 16 a. Find a formula for D b. Find D when r = 4 D  1/r T  1/n D = k/r T = k/n Sub 5 = k/16 Sub 40 = k/3 k = 5 x 4 k = 40 x 3 = 120 k = 20 D = 20/r T = 120/n D = 20/4 T = 120/10 D = 20/2 T = 12 D = 10

  36. Worked examples – Inverse Proportion ex 1 Winnings, W varies indirectly as the Syndicates, S. Given that W = 800 when S = 9 a. Find a formula for W b. Find W when S = 2 ex 2 F varies inversely as the square of d, Given that F = 10 when d = 6 a. Find a formula for F b. Find F when d = 2 W  1/S W = k/S Sub 800 = k/9 k = 800 x 9 = 7200 W = 7200/S W = 7200/2 W = 3600

  37. Worked examples – Inverse Proportion ex 3 Time, T varies indirectly as the helpers, h and the speed, s. Given that T = 4 when h = 2 and s = 6 a. Find a formula for T b. Find T when h = 3 and s = 8 Eg 1 E varies directly as m and inversely as the square of r, Given that E = 12 when m = 3 and r = 6 a. Find a formula for E b. Find E when m = 100 and r = 5 T  1/hs T = k/hs Sub 4 = k/2x6 k = 4 x 12 = 48 T = 48/hs T = 48/3x8 = 48/24 T = 2

  38. Worked examples – Joint Variation ex 1 Cost, C varies directly as the energy used, E and inversely insulation, I. Given that C = 20 when E = 60 and I = 15 a. Find a formula for E b. Find C when E = 40 and I = 25 ex 2 E varies directly as m and n inversely as the square of r, Given that a. Find a General formula for E b. What happens to E when m is halved? c. What happens to E when r is trebled? C  E/I C = kE/I Sub 20 = 60k/15 k = 15 x 20  60 = 5 E halved C = 5E/I E divided by 9 C = 5x40/25 C = 8 E ‘ninthed’

  39. Proportional division eg 1 Divide £600 in the ratio 2:1 eg 2 Divide £200 in the ratio 3:2 Total of 3 shares Total of 5 shares 2/3 2 ÷ 3 × 600 = £400 3/5 3 ÷ 5 × 200 = £120 2/5 2 ÷ 5 × 200 = £80 1/3 1 ÷ 3 × 600 = £200 ex 1 Divide £800 in the ratio 3:1 ex 2 Divide £72 in the ratio 3:5 Total of 8 shares Total of 4 shares 3/8 3 ÷ 8 × 72 = £27 3/4 3 ÷ 4 × 800 = £600 5/8 5 ÷ 8 × 72 = £45 1/4 1 ÷ 4 × 800 = £200

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